Just a bit more "data" in the form of images. I started with a "random polyhedron"
built from the convex hull of a small number of points. Then I iterated the process
of replacing each face by its centroid, i.e., $\alpha=\beta=\gamma=\frac{1}{3}$
in my original description, and
again taking the convex hull. At least visually, the data support Gjergji's suggestion
that the process converges to ellipsoids (of different axes lengths).
<br />&nbsp;&nbsp;&nbsp;![CentroidPolyhedra][1]
<br />
So, to make an explicit conjecture out of these observations:

> **Conjecture**. Given any convex polyhedron $P$ in $\mathbb{R}^3$, 
let $c(P)$ be the convex hull of the
centroids of the faces of $P$. Then $c^k(P)$ converges to an ellipsoid
as $k \to \infty$.

Secondarily, it makes sense to conjecture the same holds for any convex
polytope $P$ in $\mathbb{R}^d$.


  [1]: https://i.sstatic.net/mGkVw.jpg